When Flemming Ornskov was named chief executive of Shire PLC last year, he moved his office from the drug maker’s Dublin headquarters to its Lexington campus so he could scout for biotechs to buy here.

Now Shire itself is a takeover target. It rebuffed a $46 billion bid from pharmaceutical giant AbbVie Inc. of Chicago late last month, but the suitor hasn’t given up. It’s not only after Shire’s drug portfolio, but also the company’s address in Ireland, where corporate taxes are lower.

The AbbVie move came less than a week after Medtronic Inc. agreed to pay $42.9 billion for Covidien PLC., a supplier of health care products that bases its corporate staff and US headquarters in Mansfield. But like Shire, Covidien calls Ireland home for tax purposes.

“It’s becoming increasingly disadvantageous to be a US-based multinational,” said Eric Toder, codirector of the Tax Policy Center, a nonpartisan Washington, D.C., think tank. “So what’s the solution? You stop being a US-based multinational.”

And that’s what many American corporations are doing. Over the past decade, 40 of them have moved abroad to save money — hundreds of millions of dollars annually, in some cases — while keeping the bulk of their operations here. Covidien, for instance, has nearly 14,000 employees in the United States, including 1,800 in Mansfield, but only about 1,400 in Ireland. Shire, for its part, has about 1,500 workers in Massachusetts and just 100 in Ireland.

But there is also a backlash building in Congress and among some business leaders against businesses shopping for tax-light locales.

‘You have to run your global company for all its stakeholders. . . . These tax inversions are driving some deals. If that’s the primary rationale, you’re asking for trouble.’

Two members of the Massachusetts delegation, Senator Elizabeth Warren and Representative Richard Neal, have signed onto Democrat-sponsored bills in the Senate and the House that would tighten rules for companies that reincorporate overseas to avoid paying US taxes.

“This is one more example of Washington working for those who can afford to have armies of lobbyists and lawyers,” Warren said. “Big corporations are using the tax inversion loophole to juice their profits and avoid paying billions of dollars, while working families are forced to foot the bill.”

The bills on Capitol Hill call for a two-year moratorium on tax inversions. Michigan Democrat Senator Carl Levin, the Senate bill’s primary sponsor, describes them as “tax avoidance, plain and simple.” The legislation would also prohibit companies from shifting their tax addresses overseas if management and significant business operations remain in the United States. Republicans, who have not signed on to the bills, say it makes more sense to instead cut US corporate tax rates.

Meanwhile, at a Harvard Business School gathering last month that included chief executives from about three dozen top US companies, former Medtronic chief executive Bill George, now a Harvard management professor, called on businesses to rethink tax inversions.

”You have to run your global company for all its stakeholders . . . These tax inversions are driving some deals,” said Bill George, former Medtronic chief executive. “If that’s the primary rationale, you’re asking for trouble.”

“You have to run your global company for all its stakeholders,” George said in an interview. “That means the customers, the employees, the shareholders, and the communities you reside in. You have to be a pragmatist. These tax inversions are driving some deals. If that’s the primary rationale, you’re asking for trouble.”

George criticized pharmaceutical giant Pfizer Inc., which opened a large research center in Cambridge this month, for citing a planned tax inversion as a major reason for its $119 billion offer in April to buy London-based drug maker AstraZeneca plc, an overture that was rejected. AstraZeneca also has a Massachusetts research lab in Waltham, so had the deal gone through, the combination could have resulted in consolidation and layoffs there.

“If you’re just cutting jobs to reduce costs, that’s a one-time thing,” George said. “An acquisition has got to be good for your employees. Otherwise, you kill their motivation.”

Another growing Massachusetts biotech, Waltham-based Alkermes, bought Ireland’s Elan Drug Technologies in 2011 and promptly shifted its own headquarters to Dublin. Chief executive Richard Pops said at the time the deal wasn’t done to lower taxes — Alkermes wasn’t profitable then — but he acknowledged there would be future tax benefits.

Ireland maintains a corporate tax rate of 12.5 percent compared with the United States’s 35 percent, one of the highest rates in the world. Because of that, US companies hold about $3.5 trillion in corporate cash in offshore accounts, according to George. The money comes from sales of their products in foreign countries.

To use that cash in the United States for building plants, buying equipment, or hiring workers, companies would have to pay the 35 percent rate as a “repatriation” tax. The former Medtronic chief has called for a one-time holiday on repatriating money held abroad, but he admits it would be a temporary solution.

“Our tax rates are out of line with the rest of the world, so companies are leaving,” George said. “It’s tragic.”

Many believe a long-time solution must revolve around reforming US corporate tax policy.

“The problem is caused by US taxes being higher than everybody else’s,” said Joseph B. Darby III, a partner at Boston law firm Sullivan & Worcester. If you’re a US company, you’re a “tax prisoner,” he said.

As for the Democratic bills in Congress, few think they stand much of a chance. “I don’t think people are paying much attention to legislative proposals,” Berman said. “Congress is not in much of a position to enact tax law these days.”

Ireland also is examining ways to discourage mergers that do not involve “real substance in terms of jobs and investment in the Irish economy,” Ralph Victory, spokesman for the Irish embassy in Washington, D.C., said in an e-mail.

Covidien, formerly known as Tyco HealthCare, set up shop in Ireland shortly after former parent Tyco International spun it off in 2007. But unlike their counterparts at Pfizer, Covidien and Medtronic executives downplayed the tax benefits, saying the merger was driven by “complementary” businesses.

Neal said he spoke with Covidien chief executive Jose Almeida recently and is convinced the takeover was not based solely on tax savings.

“Seeking a favorable tax treaty is a bit different than going to Bermuda and renting a post office box,” said Neal, who in 2004 sponsored legislation that was able to stop many inversions from occurring at the time. “There is a difference between tax avoidance and tax evasion. But . . . the imminent danger here is you now have up to 50 other companies considering the same inversion process.”

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